1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 1 Equations and Inequalities

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 Quadratic Equations Solve a quadratic equation by factoring. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a quadratic equation by using the quadratic formula. Solve a quadratic equations with complex solutions. Solve applied problems. SECTION

QUADRATIC EQUATION A quadratic equation in the variable x is an equation equivalent to the equation where a, b, and c are real numbers and a ≠ 0. © 2010 Pearson Education, Inc. All rights reserved 3

THE ZERO-PRODUCT PROPERTY Let A and B be two algebraic expressions. Then AB = 0 if and only if A = 0 or B = 0. © 2010 Pearson Education, Inc. All rights reserved 4

EXAMPLE 1 Solving a Quadratic Equation by Factoring. Solve by factoring: Solution The solution set is The solutions check in the original equation. © 2010 Pearson Education, Inc. All rights reserved 5

EXAMPLE 2 Solving a Quadratic Equation by Factoring. Solve by factoring: The solution set is The solutions check in the original equation. Solution © 2010 Pearson Education, Inc. All rights reserved 6

Step 1Write the given equation in standard form so that one side is 0. Step 2Factor the nonzero side of the equation from Step 1. Step 3Set each factor obtained in Step 2 equal to 0. Step 4Solve the resulting equations in Step 3. SOLVING A QUADRATIC EQUATION BY FACTORING Step 5Check the solutions obtained in Step 4 in the original equation. © 2010 Pearson Education, Inc. All rights reserved 7

EXAMPLE 3 Solving a Quadratic Equation by Factoring. Solve by factoring: The solution set is {4}. Solution Step 1 Step 2 Check the solution in original equation. Step 5 Step 3 Step 4 © 2010 Pearson Education, Inc. All rights reserved 8

Suppose u is any algebraic expression and d ≥ 0. THE SQUARE ROOT PROPERTY © 2010 Pearson Education, Inc. All rights reserved 9

EXAMPLE 4 Solving an Equation by the Square Root Method Solve: Solution The solution set is © 2010 Pearson Education, Inc. All rights reserved 10

A quadratic trinomial x in with coefficient of x 2 equal to 1 is a perfect-square trinomial if the constant term is the square of one-half the coefficient of x. PERFECT SQUARE TRNOMIAL © 2010 Pearson Education, Inc. All rights reserved 11

EXAMPLE 5 Solving a Quadratic Equation by Completing the Square Solve by completing the square: Solution The solution set is © 2010 Pearson Education, Inc. All rights reserved 12

Step 1Rearrange the quadratic equation so that the terms in x 2 and x are on the left side of the equation and the constant term is on the right side. Step 2Make the coefficient of x 2 equal to 1 by dividing both sides of the equation by the original coefficient. (Steps 1and 2 are interchangeable.) METHOD OF COMPLETING THE SQUARE © 2010 Pearson Education, Inc. All rights reserved 13

Step 3Add the square of one-half the coefficient of x to both sides of the equation. Step 4Write the equation in the form (x + k) 2 = d using the fact that the left side is a perfect square. METHOD OF COMPLETING THE SQUARE Step 5Take the square root of each side, prefixing ± to the right side. Step 6Solve the two equations from Step 5. © 2010 Pearson Education, Inc. All rights reserved 14

EXAMPLE 6 Solving a Quadratic Equation by Completing the Square Solve by completing the square: Solution Step 1 Step 2 Step 3 © 2010 Pearson Education, Inc. All rights reserved 15

Solution continued The solution set is Step 4 Step 5 Step 6 © 2010 Pearson Education, Inc. All rights reserved 16 EXAMPLE 6 Solving a Quadratic Equation by Completing the Square

The solutions of the quadratic equation in the standard form ax 2 + bx + c = 0 with a ≠ 0 are given by the formula THE QUADRATIC FORMULA © 2010 Pearson Education, Inc. All rights reserved 17

EXAMPLE 7 Solving a Quadratic Equation by Using the Quadratic Formula Solve by using the quadratic formula. Solution © 2010 Pearson Education, Inc. All rights reserved 18

Solution continued The solution set is © 2010 Pearson Education, Inc. All rights reserved 19 EXAMPLE 7 Solving a Quadratic Equation by Using the Quadratic Formula

EXAMPLE 8 Solving Quadratic Equations by with Complex Solutions The solution set is Solve each equation. Solution © 2010 Pearson Education, Inc. All rights reserved 20

The solution set is Solution continued © 2010 Pearson Education, Inc. All rights reserved 21 EXAMPLE 8 Solving Quadratic Equations by with Complex Solutions

In the quadratic formula THE DISCRIMINANT the quantity b 2 – 4ac under the radical sign is called the discriminant of the equation. The discriminant reveals the type of solutions of the equation. © 2010 Pearson Education, Inc. All rights reserved 22

THE DISCRIMINANT DiscriminantSolutions b 2 – 4ac > 0Two unequal real b 2 – 4ac = 0One real b 2 – 4ac < 0Two nonreal complex © 2010 Pearson Education, Inc. All rights reserved 23

EXAMPLE 9 Using the Discriminant Solution Use the discriminant to determine the number and type of solutions of each quadratic equation. b 2 – 4acConclusion (–4) 2 – 4(1)(2) = 8 > 0Two unequal real 2 2 – 4(2)(19) = –148 < 0Two nonreal complex 4 2 – 4(4)(1) = 0One real © 2010 Pearson Education, Inc. All rights reserved 24

EXAMPLE 10 Partitioning a Building A rectangular building whose depth (from the front of the building) is three times its frontage is divided into two parts by a partition that is 45 feet from and parallel to the front wall. Assuming the rear portion of the building contains 2100 square feet, find the dimensions of the building. Let x = frontage of building, in feet. 3x = depth of building, in feet. 3x – 45 = depth of rear portion, in feet. Solution © 2010 Pearson Education, Inc. All rights reserved 25

EXAMPLE 10 Partitioning a Building Solution continued Area of rear = 2100 Frontage: x = 35 ft Depth: 3x = 105 ft Choose x = 35 which is > 0 © 2010 Pearson Education, Inc. All rights reserved 26

GOLDEN RECTANGLE The ration of the longer side to the shorter side of a golden rectangle is called the golden ratio. A rectangle of length p and width q, with p > q, is called a golden rectangle if you can divide the rectangle into a square with side of length q and a smaller rectangle that is similar to the original one. © 2010 Pearson Education, Inc. All rights reserved 27

EXAMPLE 11 Calculating the Golden Ratio Use the figure of the golden rectangle to calculate the golden ratio. © 2010 Pearson Education, Inc. All rights reserved 28

EXAMPLE 11 Calculating the Golden Ratio Solution © 2010 Pearson Education, Inc. All rights reserved 29

Solution © 2010 Pearson Education, Inc. All rights reserved 30 EXAMPLE 11 Calculating the Golden Ratio